双曲線関数入りの級数を計算する

こんな級数が知られているんですが、より一般的に
$$ \sum _{n=1}^\infty \frac {n^{2s}}{\sinh ^{2r}\pi n} $$
の形の級数をガンマ関数で書く方法を見つけた(と言っても既知の結果ですが)ので、書きます。特殊値も幾つか計算したので最後に掲載しておきます。
最終的に重要となるのは正規化されたアイゼンシュタイン級数

とその微分です。では計算方法を説明します。

解説

$k$を正整数として、以下の変形を考えます。
$$ \begin {aligned} \frac {d^2}{dx^2 }\frac {1}{\sinh ^{2k}\frac {x}2}&=-k\frac{d}{dx}\frac {\cosh \frac {x}2}{\sinh ^{2k+1}\frac {x}2}\\ &=-\frac {k}2\frac {1}{\sinh ^{2}\frac {x}2}+\frac {k(2k+1)}2\frac {\cosh ^{2}\frac {x}2}{\sinh ^{2k+2}\frac {x}2}\\ &=k^{2}\frac {1}{\sinh ^{2k}\frac {x}2}+\frac {k(2k+1)}2\frac {1}{\sinh ^{2(k+1)}\frac {x}2} \end {aligned} $$
これを繰り返し適用することで、$r$を正整数とするとき、定数$a_{r,1},...,a_{r,r}$を用いて
$$ \begin {aligned} \frac {d^{2(r-1)}}{dx^{2(r-1)}}\frac {1}{4\sinh ^{2}\frac {x}2}&=\sum _{k=1}^{r}a_{r,k}\frac {1}{\sinh ^{2k}\frac {x}2} \end {aligned} $$
と表せることが分かります。例えば
$$ \begin {aligned} \frac {d^2}{dx^2}\frac {1}{4\sinh ^{2}\frac {x}2}&=\frac {1}4\frac {1}{\sinh ^{2}\frac {x}2}+\frac {3}8\frac {1}{\sinh ^{4}\frac {x}2} \end {aligned} $$
となります。
以下$x>0$とします。
$$ \begin {aligned} \sum _{n=1}^\infty e^{-nx}&=\frac {1}{e^{x}-1} \end {aligned} $$
の両辺を$x$で微分して
$$ \begin {aligned} \sum _{n=1}^\infty ne^{-2nx}&=\frac {1}{4\sinh ^{2}\frac {x}2} \end {aligned} $$
を得ます。さらに両辺を$2(r-1)$回微分して
$$ \begin {aligned} \sum _{n=1}^\infty n^{2r-1}e^{-nx}&=\sum _{k=1}^{r}a_{r,k}\frac {1}{\sinh ^{2k}\frac {x}2} \end {aligned} $$
となります。ここで、
$$ \begin {aligned} T_r(x)&:=\sum _{n=1}^\infty n^{2r-1}e^{-nx} \end {aligned} $$
とおきます。$T_r$を$\operatorname{csch}^{2k}\frac{x}2$の線形結合で書くことができたので、今度は$\operatorname{csch}^{2r}\frac x2$を$T_k$の線形結合で書くことを考えます。そのために以下の行列を定義します。
$$ \begin {aligned} A&=\left [\begin{array}{ccc} a_{1,1}&&O\\ \vdots &\ddots&\\ a_{r,1}&\cdots &a_{r,r} \end{array} \right ]\\ S&=\left [\begin{array}{c} \frac {1}{\sinh ^{2}\frac {x}2}\\ \vdots \\ \frac {1}{\sinh ^{2r}\frac {x}2} \end{array} \right ], T=\left [\begin{array}{c} T_1\\ \vdots \\ T_r\end{array} \right ] \end {aligned} $$
上記より$T=AS$です。また$a_{k,k}\neq 0$ですから$|A|\neq 0$となり$A$は正則です。従って
$$ \begin {aligned} C&=\left [\begin{array}{} 0&\cdots &0&1\end{array} \right ]A^{-1}\\ &=\left [\begin{array}{} c_1&\cdots &c_r\end{array} \right ] \end {aligned} $$
とすれば
$$ \begin {aligned} CT&=\left [\begin{array}{} 0&\cdots &0&1\end{array} \right ]A^{-1}AS\\ &=\left [\begin{array}{} 0&\cdots &0&1\end{array} \right ]S\\ &=\frac {1}{\sinh ^{2r}\frac {x}2} \end {aligned} $$
となり、
$$ \begin {aligned} \frac {1}{\sinh ^{2r}\frac {x}2}&=\sum _{k=1}^{r}c_k\sum _{n=1}^\infty n^{2k-1}e^{-nx}\\ \sum _{n=1}^\infty \frac {n^{2s}}{\sinh ^{2r}\frac {nx}2} &=\sum _{k=1}^{r}c_k\sum _{n,m=1}^{\infty }n^{2k-1}m^{2s}e^{-nmx}\\ &=\sum _{k=1}^{r}c_k\left (\frac {d}{dx}\right )^{\min (2k-1,2s)}\sum _{n,m=1}^\infty n^{|2k-2s-1|}e^{-nmx}\\ &=\sum _{k=1}^{r}c_k\left (\frac {d}{dx}\right )^{\min (2k-1,2s)}\sum _{n=1}^\infty \frac {n^{|2k-2s-1|}}{e^{nx}-1}\\ &=\sum _{k=1}^{r}c_k\frac {\zeta(-1-2|2k-2s-1|)}2\left (\frac {d}{dx}\right )^{\min (2k-1,2s)}E_{|2k-2s-1|+1}\left (\frac{ix}{2\pi }\right ) \end {aligned} $$
最後の微分はこちらの記事の内容により
$E_2,E_4,E_6$で書けます。ここで$x=2\pi $とすれば
$$ \sum _{n=1}^\infty \frac {n^{2s}}{\sinh ^{2r}\pi n} $$
を
$$ \begin {aligned} E_2(i)=\frac {3}\pi ,E_4(i)=\frac {3\pi ^{2}}{4\Gamma \left (\frac {3}4\right )^8}, E_6(i)&=0 \end {aligned} $$
で書けることになります。アイゼンシュタイン級数の特殊値の求め方について補足すると、$E_2とE_6$の特殊値についてはモジュラー関係式
$$ \begin {aligned} E_2\left (-\frac {1}{\tau }\right )&=\tau ^{2}E_2(\tau )-\frac {6\tau i}\pi \\ E_6\left (-\frac {1}\tau \right )&=\tau ^{6}E_6(\tau ) \end {aligned} $$
において$\tau =i$とすれば求まり、$E_4$については
この記事で証明されている
$$ \begin {aligned} \eta(\tau )^{24}&=\frac {E_4(\tau)^3-E_6(\tau )^2}{1728} \end {aligned} $$
において$\tau = i$とし、この記事で解説したイータ関数の特殊値
$$ \begin {aligned} \eta(i)&=\frac {\sqrt [4]\pi }{\sqrt 2\Gamma \left (\frac {3}4\right )} \end {aligned} $$
を用いて求まります。

発展(Help!)

実は、途中出てきた$a_{r,k}$の明示式
$$ \begin {aligned} a_{r,k}&=\frac {(2k-1)!}{2^{2k}}\sum _{1=n_0\leq n_1\leq \cdots \leq n_{r-k}\leq k}n_0^2\cdots n_{r-k}^2 \end {aligned} $$
を得たんですが、使い道が見つからず困っています。うまく多重ゼータ値の話を持ち込めたりしないかな、という気持ちもあるので、何か知っている方は是非教えてください～

特殊値鑑賞会

プログラムを使って特殊値を大量に計算してみました。以下では定数$r$を
$$ \begin {aligned} r&=\frac {\pi ^{4}}{4\Gamma \left (\frac {3}4\right )^8} \end{aligned} $$
として使います。
$$ \begin {align*}\sum _{n=1}^\infty \frac {1}{\sinh ^{2}\pi n}&=\frac{1}{6}-\frac{1}{2}\pi ^{-1}\\ \sum _{n=1}^\infty \frac {n^{2}}{\sinh ^{2}\pi n}&=-\frac{1}{8}\pi ^{-2}+\frac{1}{24}\pi ^{-2}r\\ \sum _{n=1}^\infty \frac {n^{4}}{\sinh ^{2}\pi n}&=\frac{1}{20}\pi ^{-3}r\\ \sum _{n=1}^\infty \frac {n^{6}}{\sinh ^{2}\pi n}&=\frac{1}{28}\pi ^{-4}r^{2}\\ \sum _{n=1}^\infty \frac {n^{8}}{\sinh ^{2}\pi n}&=\frac{3}{20}\pi ^{-5}r^{2}\\ \sum _{n=1}^\infty \frac {n^{10}}{\sinh ^{2}\pi n}&=\frac{9}{44}\pi ^{-6}r^{3}\\ \sum _{n=1}^\infty \frac {n^{12}}{\sinh ^{2}\pi n}&=\frac{567}{260}\pi ^{-7}r^{3}\\ \sum _{n=1}^\infty \frac {n^{14}}{\sinh ^{2}\pi n}&=-\frac{81}{8}\pi ^{-6}r^{2}+\frac{3}{8}\pi ^{-6}r^{3}\\ \end {align*} $$
$$ \begin {align*}\sum _{n=1}^\infty \frac {1}{\sinh ^{4}\pi n}&=-\frac{11}{90}+\frac{1}{3}\pi ^{-1}+\frac{1}{30}\pi ^{-2}r\\ \sum _{n=1}^\infty \frac {n^{2}}{\sinh ^{4}\pi n}&=\frac{1}{12}\pi ^{-2}-\frac{1}{36}\pi ^{-2}r-\frac{1}{24}\pi ^{-3}+\frac{1}{24}\pi ^{-3}r\\ \sum _{n=1}^\infty \frac {n^{4}}{\sinh ^{4}\pi n}&=-\frac{1}{30}\pi ^{-3}r-\frac{1}{32}\pi ^{-4}+\frac{1}{16}\pi ^{-4}r+\frac{1}{96}\pi ^{-4}r^{2}\\ \sum _{n=1}^\infty \frac {n^{6}}{\sinh ^{4}\pi n}&=-\frac{1}{42}\pi ^{-4}r^{2}+\frac{1}{16}\pi ^{-5}r+\frac{1}{16}\pi ^{-5}r^{2}\\ \sum _{n=1}^\infty \frac {n^{8}}{\sinh ^{4}\pi n}&=-\frac{1}{10}\pi ^{-5}r^{2}+\frac{1}{4}\pi ^{-6}r^{2}+\frac{1}{72}\pi ^{-6}r^{3}\\ \sum _{n=1}^\infty \frac {n^{10}}{\sinh ^{4}\pi n}&=-\frac{3}{22}\pi ^{-6}r^{3}+\frac{9}{16}\pi ^{-7}r^{2}+\frac{5}{16}\pi ^{-7}r^{3}\\ \sum _{n=1}^\infty \frac {n^{12}}{\sinh ^{4}\pi n}&=-\frac{189}{130}\pi ^{-7}r^{3}+\frac{27}{8}\pi ^{-8}r^{3}+\frac{1}{4}\pi ^{-8}r^{4}\\ \sum _{n=1}^\infty \frac {n^{14}}{\sinh ^{4}\pi n}&=\frac{27}{4}\pi ^{-6}r^{2}-\frac{1}{4}\pi ^{-6}r^{3}+\frac{1323}{80}\pi ^{-9}r^{3}+\frac{171}{16}\pi ^{-9}r^{4}\\ \end {align*} $$
$$ \begin {align*}\sum _{n=1}^\infty \frac {1}{\sinh ^{6}\pi n}&=\frac{191}{1890}-\frac{4}{15}\pi ^{-1}-\frac{1}{30}\pi ^{-2}r\\ \sum _{n=1}^\infty \frac {n^{2}}{\sinh ^{6}\pi n}&=-\frac{1}{15}\pi ^{-2}+\frac{1}{45}\pi ^{-2}r+\frac{1}{24}\pi ^{-3}-\frac{1}{24}\pi ^{-3}r+\frac{1}{120}\pi ^{-4}r+\frac{1}{360}\pi ^{-4}r^{2}\\ \sum _{n=1}^\infty \frac {n^{4}}{\sinh ^{6}\pi n}&=\frac{2}{75}\pi ^{-3}r+\frac{1}{32}\pi ^{-4}-\frac{1}{16}\pi ^{-4}r-\frac{1}{96}\pi ^{-4}r^{2}-\frac{1}{160}\pi ^{-5}+\frac{1}{48}\pi ^{-5}r+\frac{1}{96}\pi ^{-5}r^{2}\\ \sum _{n=1}^\infty \frac {n^{6}}{\sinh ^{6}\pi n}&=\frac{2}{105}\pi ^{-4}r^{2}-\frac{1}{16}\pi ^{-5}r-\frac{1}{16}\pi ^{-5}r^{2}-\frac{1}{128}\pi ^{-6}+\frac{5}{128}\pi ^{-6}r+\frac{5}{128}\pi ^{-6}r^{2}+\frac{1}{1920}\pi ^{-6}r^{3}\\ \sum _{n=1}^\infty \frac {n^{8}}{\sinh ^{6}\pi n}&=\frac{2}{25}\pi ^{-5}r^{2}-\frac{1}{4}\pi ^{-6}r^{2}-\frac{1}{72}\pi ^{-6}r^{3}+\frac{7}{160}\pi ^{-7}r+\frac{7}{48}\pi ^{-7}r^{2}+\frac{1}{96}\pi ^{-7}r^{3}\\ \sum _{n=1}^\infty \frac {n^{10}}{\sinh ^{6}\pi n}&=\frac{6}{55}\pi ^{-6}r^{3}-\frac{9}{16}\pi ^{-7}r^{2}-\frac{5}{16}\pi ^{-7}r^{3}+\frac{15}{32}\pi ^{-8}r^{2}+\frac{5}{32}\pi ^{-8}r^{3}+\frac{1}{240}\pi ^{-8}r^{4}\\ \sum _{n=1}^\infty \frac {n^{12}}{\sinh ^{6}\pi n}&=\frac{378}{325}\pi ^{-7}r^{3}-\frac{27}{8}\pi ^{-8}r^{3}-\frac{1}{4}\pi ^{-8}r^{4}+\frac{297}{320}\pi ^{-9}r^{2}+\frac{55}{32}\pi ^{-9}r^{3}+\frac{13}{64}\pi ^{-9}r^{4}\\ \sum _{n=1}^\infty \frac {n^{14}}{\sinh ^{6}\pi n}&=-\frac{27}{5}\pi ^{-6}r^{2}+\frac{1}{5}\pi ^{-6}r^{3}-\frac{1323}{80}\pi ^{-9}r^{3}-\frac{171}{16}\pi ^{-9}r^{4}+\frac{819}{64}\pi ^{-10}r^{3}+\frac{91}{16}\pi ^{-10}r^{4}+\frac{107}{960}\pi ^{-10}r^{5}\\ \end {align*} $$
$$ \begin {align*}\sum _{n=1}^\infty \frac {1}{\sinh ^{8}\pi n}&=-\frac{2497}{28350}+\frac{8}{35}\pi ^{-1}+\frac{7}{225}\pi ^{-2}r+\frac{1}{1050}\pi ^{-4}r^{2}\\ \sum _{n=1}^\infty \frac {n^{2}}{\sinh ^{8}\pi n}&=\frac{2}{35}\pi ^{-2}-\frac{2}{105}\pi ^{-2}r-\frac{7}{180}\pi ^{-3}+\frac{7}{180}\pi ^{-3}r-\frac{1}{90}\pi ^{-4}r-\frac{1}{270}\pi ^{-4}r^{2}+\frac{1}{630}\pi ^{-5}r^{2}\\ \sum _{n=1}^\infty \frac {n^{4}}{\sinh ^{8}\pi n}&=-\frac{4}{175}\pi ^{-3}r-\frac{7}{240}\pi ^{-4}+\frac{7}{120}\pi ^{-4}r+\frac{7}{720}\pi ^{-4}r^{2}+\frac{1}{120}\pi ^{-5}-\frac{1}{36}\pi ^{-5}r-\frac{1}{72}\pi ^{-5}r^{2}+\frac{1}{480}\pi ^{-6}r+\frac{1}{240}\pi ^{-6}r^{2}+\frac{1}{10080}\pi ^{-6}r^{3}\\ \sum _{n=1}^\infty \frac {n^{6}}{\sinh ^{8}\pi n}&=-\frac{4}{245}\pi ^{-4}r^{2}+\frac{7}{120}\pi ^{-5}r+\frac{7}{120}\pi ^{-5}r^{2}+\frac{1}{96}\pi ^{-6}-\frac{5}{96}\pi ^{-6}r-\frac{5}{96}\pi ^{-6}r^{2}-\frac{1}{1440}\pi ^{-6}r^{3}-\frac{1}{896}\pi ^{-7}+\frac{1}{128}\pi ^{-7}r+\frac{5}{384}\pi ^{-7}r^{2}+\frac{1}{1920}\pi ^{-7}r^{3}\\ \sum _{n=1}^\infty \frac {n^{8}}{\sinh ^{8}\pi n}&=-\frac{12}{175}\pi ^{-5}r^{2}+\frac{7}{30}\pi ^{-6}r^{2}+\frac{7}{540}\pi ^{-6}r^{3}-\frac{7}{120}\pi ^{-7}r-\frac{7}{36}\pi ^{-7}r^{2}-\frac{1}{72}\pi ^{-7}r^{3}-\frac{1}{512}\pi ^{-8}+\frac{7}{384}\pi ^{-8}r+\frac{35}{768}\pi ^{-8}r^{2}+\frac{7}{1920}\pi ^{-8}r^{3}+\frac{13}{161280}\pi ^{-8}r^{4}\\ \sum _{n=1}^\infty \frac {n^{10}}{\sinh ^{8}\pi n}&=-\frac{36}{385}\pi ^{-6}r^{3}+\frac{21}{40}\pi ^{-7}r^{2}+\frac{7}{24}\pi ^{-7}r^{3}-\frac{5}{8}\pi ^{-8}r^{2}-\frac{5}{24}\pi ^{-8}r^{3}-\frac{1}{180}\pi ^{-8}r^{4}+\frac{3}{128}\pi ^{-9}r+\frac{21}{128}\pi ^{-9}r^{2}+\frac{5}{128}\pi ^{-9}r^{3}+\frac{1}{384}\pi ^{-9}r^{4}\\ \sum _{n=1}^\infty \frac {n^{12}}{\sinh ^{8}\pi n}&=-\frac{324}{325}\pi ^{-7}r^{3}+\frac{63}{20}\pi ^{-8}r^{3}+\frac{7}{30}\pi ^{-8}r^{4}-\frac{99}{80}\pi ^{-9}r^{2}-\frac{55}{24}\pi ^{-9}r^{3}-\frac{13}{48}\pi ^{-9}r^{4}+\frac{33}{64}\pi ^{-10}r^{2}+\frac{55}{128}\pi ^{-10}r^{3}+\frac{11}{160}\pi ^{-10}r^{4}+\frac{1}{1920}\pi ^{-10}r^{5}\\ \sum _{n=1}^\infty \frac {n^{14}}{\sinh ^{8}\pi n}&=\frac{162}{35}\pi ^{-6}r^{2}-\frac{6}{35}\pi ^{-6}r^{3}+\frac{3087}{200}\pi ^{-9}r^{3}+\frac{399}{40}\pi ^{-9}r^{4}-\frac{273}{16}\pi ^{-10}r^{3}-\frac{91}{12}\pi ^{-10}r^{4}-\frac{107}{720}\pi ^{-10}r^{5}+\frac{1287}{1280}\pi ^{-11}r^{2}+\frac{1001}{256}\pi ^{-11}r^{3}+\frac{1183}{768}\pi ^{-11}r^{4}+\frac{217}{3840}\pi ^{-11}r^{5}\\ \end {align*} $$
$$ \begin {align*}\sum _{n=1}^\infty \frac {1}{\sinh ^{10}\pi n}&=\frac{14797}{187110}-\frac{64}{315}\pi ^{-1}-\frac{82}{2835}\pi ^{-2}r-\frac{1}{630}\pi ^{-4}r^{2}\\ \sum _{n=1}^\infty \frac {n^{2}}{\sinh ^{10}\pi n}&=-\frac{16}{315}\pi ^{-2}+\frac{16}{945}\pi ^{-2}r+\frac{41}{1134}\pi ^{-3}-\frac{41}{1134}\pi ^{-3}r+\frac{13}{1080}\pi ^{-4}r+\frac{13}{3240}\pi ^{-4}r^{2}-\frac{1}{378}\pi ^{-5}r^{2}+\frac{1}{4200}\pi ^{-6}r^{2}+\frac{1}{22680}\pi ^{-6}r^{3}\\ \sum _{n=1}^\infty \frac {n^{4}}{\sinh ^{10}\pi n}&=\frac{32}{1575}\pi ^{-3}r+\frac{41}{1512}\pi ^{-4}-\frac{41}{756}\pi ^{-4}r-\frac{41}{4536}\pi ^{-4}r^{2}-\frac{13}{1440}\pi ^{-5}+\frac{13}{432}\pi ^{-5}r+\frac{13}{864}\pi ^{-5}r^{2}-\frac{1}{288}\pi ^{-6}r-\frac{1}{144}\pi ^{-6}r^{2}-\frac{1}{6048}\pi ^{-6}r^{3}+\frac{1}{1260}\pi ^{-7}r^{2}+\frac{1}{7560}\pi ^{-7}r^{3}\\ \sum _{n=1}^\infty \frac {n^{6}}{\sinh ^{10}\pi n}&=\frac{32}{2205}\pi ^{-4}r^{2}-\frac{41}{756}\pi ^{-5}r-\frac{41}{756}\pi ^{-5}r^{2}-\frac{13}{1152}\pi ^{-6}+\frac{65}{1152}\pi ^{-6}r+\frac{65}{1152}\pi ^{-6}r^{2}+\frac{13}{17280}\pi ^{-6}r^{3}+\frac{5}{2688}\pi ^{-7}-\frac{5}{384}\pi ^{-7}r-\frac{25}{1152}\pi ^{-7}r^{2}-\frac{1}{1152}\pi ^{-7}r^{3}+\frac{1}{1920}\pi ^{-8}r+\frac{1}{384}\pi ^{-8}r^{2}+\frac{1}{2688}\pi ^{-8}r^{3}+\frac{1}{120960}\pi ^{-8}r^{4}\\ \sum _{n=1}^\infty \frac {n^{8}}{\sinh ^{10}\pi n}&=\frac{32}{525}\pi ^{-5}r^{2}-\frac{41}{189}\pi ^{-6}r^{2}-\frac{41}{3402}\pi ^{-6}r^{3}+\frac{91}{1440}\pi ^{-7}r+\frac{91}{432}\pi ^{-7}r^{2}+\frac{13}{864}\pi ^{-7}r^{3}+\frac{5}{1536}\pi ^{-8}-\frac{35}{1152}\pi ^{-8}r-\frac{175}{2304}\pi ^{-8}r^{2}-\frac{7}{1152}\pi ^{-8}r^{3}-\frac{13}{96768}\pi ^{-8}r^{4}-\frac{1}{4608}\pi ^{-9}+\frac{1}{384}\pi ^{-9}r+\frac{7}{768}\pi ^{-9}r^{2}+\frac{7}{5760}\pi ^{-9}r^{3}+\frac{13}{161280}\pi ^{-9}r^{4}\\ \sum _{n=1}^\infty \frac {n^{10}}{\sinh ^{10}\pi n}&=\frac{32}{385}\pi ^{-6}r^{3}-\frac{41}{84}\pi ^{-7}r^{2}-\frac{205}{756}\pi ^{-7}r^{3}+\frac{65}{96}\pi ^{-8}r^{2}+\frac{65}{288}\pi ^{-8}r^{3}+\frac{13}{2160}\pi ^{-8}r^{4}-\frac{5}{128}\pi ^{-9}r-\frac{35}{128}\pi ^{-9}r^{2}-\frac{25}{384}\pi ^{-9}r^{3}-\frac{5}{1152}\pi ^{-9}r^{4}-\frac{1}{2048}\pi ^{-10}+\frac{15}{2048}\pi ^{-10}r+\frac{35}{1024}\pi ^{-10}r^{2}+\frac{7}{1024}\pi ^{-10}r^{3}+\frac{13}{14336}\pi ^{-10}r^{4}+\frac{1}{645120}\pi ^{-10}r^{5}\\ \sum _{n=1}^\infty \frac {n^{12}}{\sinh ^{10}\pi n}&=\frac{288}{325}\pi ^{-7}r^{3}-\frac{41}{14}\pi ^{-8}r^{3}-\frac{41}{189}\pi ^{-8}r^{4}+\frac{429}{320}\pi ^{-9}r^{2}+\frac{715}{288}\pi ^{-9}r^{3}+\frac{169}{576}\pi ^{-9}r^{4}-\frac{55}{64}\pi ^{-10}r^{2}-\frac{275}{384}\pi ^{-10}r^{3}-\frac{11}{96}\pi ^{-10}r^{4}-\frac{1}{1152}\pi ^{-10}r^{5}+\frac{11}{1024}\pi ^{-11}r+\frac{33}{256}\pi ^{-11}r^{2}+\frac{33}{512}\pi ^{-11}r^{3}+\frac{11}{768}\pi ^{-11}r^{4}+\frac{19}{107520}\pi ^{-11}r^{5}\\ \sum _{n=1}^\infty \frac {n^{14}}{\sinh ^{10}\pi n}&=-\frac{144}{35}\pi ^{-6}r^{2}+\frac{16}{105}\pi ^{-6}r^{3}-\frac{287}{20}\pi ^{-9}r^{3}-\frac{779}{84}\pi ^{-9}r^{4}+\frac{1183}{64}\pi ^{-10}r^{3}+\frac{1183}{144}\pi ^{-10}r^{4}+\frac{1391}{8640}\pi ^{-10}r^{5}-\frac{429}{256}\pi ^{-11}r^{2}-\frac{5005}{768}\pi ^{-11}r^{3}-\frac{5915}{2304}\pi ^{-11}r^{4}-\frac{217}{2304}\pi ^{-11}r^{5}+\frac{429}{1024}\pi ^{-12}r^{2}+\frac{1001}{1536}\pi ^{-12}r^{3}+\frac{1001}{3840}\pi ^{-12}r^{4}+\frac{91}{7680}\pi ^{-12}r^{5}+\frac{29}{322560}\pi ^{-12}r^{6}\\ \end {align*} $$
$$ \begin {align*}\sum _{n=1}^\infty \frac {1}{\sinh ^{12}\pi n}&=-\frac{92427157}{1277025750}+\frac{128}{693}\pi ^{-1}+\frac{1916}{70875}\pi ^{-2}r+\frac{31}{15750}\pi ^{-4}r^{2}+\frac{1}{53625}\pi ^{-6}r^{3}\\ \sum _{n=1}^\infty \frac {n^{2}}{\sinh ^{12}\pi n}&=\frac{32}{693}\pi ^{-2}-\frac{32}{2079}\pi ^{-2}r-\frac{479}{14175}\pi ^{-3}+\frac{479}{14175}\pi ^{-3}r-\frac{139}{11340}\pi ^{-4}r-\frac{139}{34020}\pi ^{-4}r^{2}+\frac{31}{9450}\pi ^{-5}r^{2}-\frac{1}{2100}\pi ^{-6}r^{2}-\frac{1}{11340}\pi ^{-6}r^{3}+\frac{1}{34650}\pi ^{-7}r^{3}\\ \sum _{n=1}^\infty \frac {n^{4}}{\sinh ^{12}\pi n}&=-\frac{64}{3465}\pi ^{-3}r-\frac{479}{18900}\pi ^{-4}+\frac{479}{9450}\pi ^{-4}r+\frac{479}{56700}\pi ^{-4}r^{2}+\frac{139}{15120}\pi ^{-5}-\frac{139}{4536}\pi ^{-5}r-\frac{139}{9072}\pi ^{-5}r^{2}+\frac{31}{7200}\pi ^{-6}r+\frac{31}{3600}\pi ^{-6}r^{2}+\frac{31}{151200}\pi ^{-6}r^{3}-\frac{1}{630}\pi ^{-7}r^{2}-\frac{1}{3780}\pi ^{-7}r^{3}+\frac{1}{16800}\pi ^{-8}r^{2}+\frac{1}{15120}\pi ^{-8}r^{3}+\frac{13}{4989600}\pi ^{-8}r^{4}\\ \sum _{n=1}^\infty \frac {n^{6}}{\sinh ^{12}\pi n}&=-\frac{64}{4851}\pi ^{-4}r^{2}+\frac{479}{9450}\pi ^{-5}r+\frac{479}{9450}\pi ^{-5}r^{2}+\frac{139}{12096}\pi ^{-6}-\frac{695}{12096}\pi ^{-6}r-\frac{695}{12096}\pi ^{-6}r^{2}-\frac{139}{181440}\pi ^{-6}r^{3}-\frac{31}{13440}\pi ^{-7}+\frac{31}{1920}\pi ^{-7}r+\frac{31}{1152}\pi ^{-7}r^{2}+\frac{31}{28800}\pi ^{-7}r^{3}-\frac{1}{960}\pi ^{-8}r-\frac{1}{192}\pi ^{-8}r^{2}-\frac{1}{1344}\pi ^{-8}r^{3}-\frac{1}{60480}\pi ^{-8}r^{4}+\frac{1}{3360}\pi ^{-9}r^{2}+\frac{1}{6048}\pi ^{-9}r^{3}+\frac{1}{75600}\pi ^{-9}r^{4}\\ \sum _{n=1}^\infty \frac {n^{8}}{\sinh ^{12}\pi n}&=-\frac{64}{1155}\pi ^{-5}r^{2}+\frac{958}{4725}\pi ^{-6}r^{2}+\frac{479}{42525}\pi ^{-6}r^{3}-\frac{139}{2160}\pi ^{-7}r-\frac{139}{648}\pi ^{-7}r^{2}-\frac{139}{9072}\pi ^{-7}r^{3}-\frac{31}{7680}\pi ^{-8}+\frac{217}{5760}\pi ^{-8}r+\frac{217}{2304}\pi ^{-8}r^{2}+\frac{217}{28800}\pi ^{-8}r^{3}+\frac{403}{2419200}\pi ^{-8}r^{4}+\frac{1}{2304}\pi ^{-9}-\frac{1}{192}\pi ^{-9}r-\frac{7}{384}\pi ^{-9}r^{2}-\frac{7}{2880}\pi ^{-9}r^{3}-\frac{13}{80640}\pi ^{-9}r^{4}+\frac{1}{7680}\pi ^{-10}r+\frac{7}{5760}\pi ^{-10}r^{2}+\frac{1}{2304}\pi ^{-10}r^{3}+\frac{1}{17280}\pi ^{-10}r^{4}+\frac{19}{79833600}\pi ^{-10}r^{5}\\ \sum _{n=1}^\infty \frac {n^{10}}{\sinh ^{12}\pi n}&=-\frac{64}{847}\pi ^{-6}r^{3}+\frac{479}{1050}\pi ^{-7}r^{2}+\frac{479}{1890}\pi ^{-7}r^{3}-\frac{695}{1008}\pi ^{-8}r^{2}-\frac{695}{3024}\pi ^{-8}r^{3}-\frac{139}{22680}\pi ^{-8}r^{4}+\frac{31}{640}\pi ^{-9}r+\frac{217}{640}\pi ^{-9}r^{2}+\frac{31}{384}\pi ^{-9}r^{3}+\frac{31}{5760}\pi ^{-9}r^{4}+\frac{1}{1024}\pi ^{-10}-\frac{15}{1024}\pi ^{-10}r-\frac{35}{512}\pi ^{-10}r^{2}-\frac{7}{512}\pi ^{-10}r^{3}-\frac{13}{7168}\pi ^{-10}r^{4}-\frac{1}{322560}\pi ^{-10}r^{5}-\frac{1}{22528}\pi ^{-11}+\frac{5}{6144}\pi ^{-11}r+\frac{5}{1024}\pi ^{-11}r^{2}+\frac{7}{5120}\pi ^{-11}r^{3}+\frac{13}{43008}\pi ^{-11}r^{4}+\frac{1}{645120}\pi ^{-11}r^{5}\\ \sum _{n=1}^\infty \frac {n^{12}}{\sinh ^{12}\pi n}&=-\frac{576}{715}\pi ^{-7}r^{3}+\frac{479}{175}\pi ^{-8}r^{3}+\frac{958}{4725}\pi ^{-8}r^{4}-\frac{1529}{1120}\pi ^{-9}r^{2}-\frac{7645}{3024}\pi ^{-9}r^{3}-\frac{1807}{6048}\pi ^{-9}r^{4}+\frac{341}{320}\pi ^{-10}r^{2}+\frac{341}{384}\pi ^{-10}r^{3}+\frac{341}{2400}\pi ^{-10}r^{4}+\frac{31}{28800}\pi ^{-10}r^{5}-\frac{11}{512}\pi ^{-11}r-\frac{33}{128}\pi ^{-11}r^{2}-\frac{33}{256}\pi ^{-11}r^{3}-\frac{11}{384}\pi ^{-11}r^{4}-\frac{19}{53760}\pi ^{-11}r^{5}-\frac{1}{8192}\pi ^{-12}+\frac{11}{4096}\pi ^{-12}r+\frac{165}{8192}\pi ^{-12}r^{2}+\frac{77}{10240}\pi ^{-12}r^{3}+\frac{143}{57344}\pi ^{-12}r^{4}+\frac{11}{430080}\pi ^{-12}r^{5}+\frac{149}{425779200}\pi ^{-12}r^{6}\\ \sum _{n=1}^\infty \frac {n^{14}}{\sinh ^{12}\pi n}&=\frac{288}{77}\pi ^{-6}r^{2}-\frac{32}{231}\pi ^{-6}r^{3}+\frac{3353}{250}\pi ^{-9}r^{3}+\frac{9101}{1050}\pi ^{-9}r^{4}-\frac{1807}{96}\pi ^{-10}r^{3}-\frac{1807}{216}\pi ^{-10}r^{4}-\frac{14873}{90720}\pi ^{-10}r^{5}+\frac{13299}{6400}\pi ^{-11}r^{2}+\frac{31031}{3840}\pi ^{-11}r^{3}+\frac{36673}{11520}\pi ^{-11}r^{4}+\frac{6727}{57600}\pi ^{-11}r^{5}-\frac{429}{512}\pi ^{-12}r^{2}-\frac{1001}{768}\pi ^{-12}r^{3}-\frac{1001}{1920}\pi ^{-12}r^{4}-\frac{91}{3840}\pi ^{-12}r^{5}-\frac{29}{161280}\pi ^{-12}r^{6}+\frac{91}{20480}\pi ^{-13}r+\frac{1001}{12288}\pi ^{-13}r^{2}+\frac{143}{2048}\pi ^{-13}r^{3}+\frac{1001}{30720}\pi ^{-13}r^{4}+\frac{247}{184320}\pi ^{-13}r^{5}+\frac{31}{921600}\pi ^{-13}r^{6}\\ \end {align*} $$
$$ \begin {align*}\sum _{n=1}^\infty \frac {1}{\sinh ^{14}\pi n}&=\frac{8113594822600175}{120861989803434770}-\frac{2379130053706055}{13954155373592348}\pi ^{-1}-\frac{887052031829709}{34885388433980870}\pi ^{-2}r-\frac{4898474577706915}{2232664859774775800}\pi ^{-4}r^{2}-\frac{169182581596875}{57156220410234260000}\pi ^{-5}r^{2}-\frac{3991455721378125}{91732205596672260000}\pi ^{-6}r^{3}\\ \sum _{n=1}^\infty \frac {n^{2}}{\sinh ^{14}\pi n}&=-\frac{7137390161118165}{167449864483108200}\pi ^{-2}+\frac{2379130053706055}{167449864483108200}\pi ^{-2}r+\frac{7983468286467381}{251174796724662270}\pi ^{-3}-\frac{2661156095489127}{83724932241554100}\pi ^{-3}r+\frac{6119155541604943}{502349593449324540}\pi ^{-4}r+\frac{8158874055473257}{2009398373797298200}\pi ^{-4}r^{2}-\frac{95424829435849}{26096082776588290}\pi ^{-5}r^{2}+\frac{2976886479699701}{4465329719549551600}\pi ^{-6}r^{2}+\frac{360834724812085}{2922761270977888000}\pi ^{-6}r^{3}-\frac{5773355596993359}{85734330615351400000}\pi ^{-7}r^{3}+\frac{2664625660150781}{571562204102342600000}\pi ^{-8}r^{3}+\frac{1928681430204375}{1.9204490057838714e+21}\pi ^{-8}r^{4}\\ \sum _{n=1}^\infty \frac {n^{4}}{\sinh ^{14}\pi n}&=\frac{7137390161118165}{418624661207770500}\pi ^{-3}r+\frac{748450151856317}{31396849590582784}\pi ^{-4}-\frac{7983468286467381}{167449864483108200}\pi ^{-4}r-\frac{2661156095489127}{334899728966216400}\pi ^{-4}r^{2}-\frac{78227840730745}{8562777161068032}\pi ^{-5}+\frac{8741650773721347}{287056910542471170}\pi ^{-5}r+\frac{8741650773721347}{574113821084942340}\pi ^{-5}r^{2}-\frac{657536715331397}{137004434577088510}\pi ^{-6}r-\frac{688847987490035}{71764227635617790}\pi ^{-6}r^{2}-\frac{95424829435849}{417537324425412600}\pi ^{-6}r^{3}+\frac{8930659439099103}{4018796747594596400}\pi ^{-7}r^{2}+\frac{2976886479699701}{8037593495189193000}\pi ^{-7}r^{3}-\frac{7422885767562891}{53444777526452810000}\pi ^{-8}r^{2}-\frac{620184683270771}{4018796747594596400}\pi ^{-8}r^{3}-\frac{2599259662715625}{427558220211622500000}\pi ^{-8}r^{4}+\frac{771208910915625}{53444777526452810000}\pi ^{-9}r^{3}+\frac{85689878990625}{26722388763226407000}\pi ^{-9}r^{4}\\ \sum _{n=1}^\infty \frac {n^{6}}{\sinh ^{14}\pi n}&=\frac{1427478032223633}{117214905138175740}\pi ^{-4}r^{2}-\frac{748450151856317}{15698424795291392}\pi ^{-5}r-\frac{7983468286467381}{167449864483108200}\pi ^{-5}r^{2}-\frac{2458589280109129}{215292682906853380}\pi ^{-6}+\frac{1792721350079573}{31396849590582784}\pi ^{-6}r+\frac{1792721350079573}{31396849590582784}\pi ^{-6}r^{2}+\frac{6119155541604943}{8037593495189193000}\pi ^{-6}r^{3}+\frac{2906077447223585}{1130286585260980200}\pi ^{-7}-\frac{352251811784677}{19572062082441216}\pi ^{-7}r-\frac{5650706147379193}{188381097543496700}\pi ^{-7}r^{2}-\frac{688847987490035}{574113821084942340}\pi ^{-7}r^{3}+\frac{2930372628454393}{2009398373797298200}\pi ^{-8}r+\frac{457870723195999}{62793699181165570}\pi ^{-8}r^{2}+\frac{1395415537359235}{1339598915864865500}\pi ^{-8}r^{3}+\frac{2976886479699701}{128601495923027080000}\pi ^{-8}r^{4}-\frac{8372493224155409}{12056390242783790000}\pi ^{-9}r^{2}-\frac{4651385124530783}{12056390242783790000}\pi ^{-9}r^{3}-\frac{6598120682278125}{213779110105811260000}\pi ^{-9}r^{4}+\frac{3181236757526953}{213779110105811260000}\pi ^{-10}r^{2}+\frac{5980352302968149}{144676682913405470000}\pi ^{-10}r^{3}+\frac{4177381600792969}{427558220211622500000}\pi ^{-10}r^{4}+\frac{204337403746875}{1.71023288084649e+21}\pi ^{-10}r^{5}\\ \sum _{n=1}^\infty \frac {n^{8}}{\sinh ^{14}\pi n}&=\frac{5353042620838623}{104656165301942620}\pi ^{-5}r^{2}-\frac{748450151856317}{3924606198822848}\pi ^{-6}r^{2}-\frac{2661156095489127}{251174796724662270}\pi ^{-6}r^{3}+\frac{8031391648356487}{125587398362331140}\pi ^{-7}r+\frac{7529429670334207}{35321455789405630}\pi ^{-7}r^{2}+\frac{8741650773721347}{574113821084942340}\pi ^{-7}r^{3}+\frac{5547966035608663}{1233039911193796600}\pi ^{-8}-\frac{8899862182122229}{211928734736433800}\pi ^{-8}r-\frac{3708275909217595}{35321455789405630}\pi ^{-8}r^{2}-\frac{263699620211029}{31396849590582784}\pi ^{-8}r^{3}-\frac{5970015891580303}{32150373980756770000}\pi ^{-8}r^{4}-\frac{3090627381572993}{5086289633674411000}\pi ^{-9}+\frac{457870723195999}{62793699181165570}\pi ^{-9}r+\frac{4807642593557989}{188381097543496700}\pi ^{-9}r^{2}+\frac{1282038024948797}{376762195086993400}\pi ^{-9}r^{3}+\frac{518297199590573}{2296455284339769300}\pi ^{-9}r^{4}-\frac{2996972006373811}{9864319289550373000}\pi ^{-10}r-\frac{6410190124743985}{2260573170521960400}\pi ^{-10}r^{2}-\frac{3329968895970901}{3288106429850124300}\pi ^{-10}r^{3}-\frac{3551966822368961}{26304851438800994000}\pi ^{-10}r^{4}-\frac{2570947414286105}{4.629653853228975e+21}\pi ^{-10}r^{5}+\frac{2651030631272461}{26722388763226407000}\pi ^{-11}r^{2}+\frac{523280826509713}{4521146341043921000}\pi ^{-11}r^{3}+\frac{6598120682278125}{213779110105811260000}\pi ^{-11}r^{4}+\frac{599829152934375}{855116440423245000000}\pi ^{-11}r^{5}\\ \sum _{n=1}^\infty \frac {n^{10}}{\sinh ^{14}\pi n}&=\frac{5353042620838623}{76747854554757920}\pi ^{-6}r^{3}-\frac{2245350455568951}{5232808265097131}\pi ^{-7}r^{2}-\frac{3742250759281585}{15698424795291392}\pi ^{-7}r^{3}+\frac{576231862525577}{840987042604896}\pi ^{-8}r^{2}+\frac{1792721350079573}{7849212397645696}\pi ^{-8}r^{3}+\frac{655623808029101}{107646341453426690}\pi ^{-8}r^{4}-\frac{5547966035608663}{102753325932816380}\pi ^{-9}r-\frac{8899862182122229}{23547637192937090}\pi ^{-9}r^{2}-\frac{1210865603009827}{13455792681678336}\pi ^{-9}r^{3}-\frac{3287683576656985}{548017738308354050}\pi ^{-9}r^{4}-\frac{2317970536179745}{1695429877891470300}\pi ^{-10}+\frac{21}{1024}\pi ^{-10}r+\frac{4507164931460615}{47095274385874180}\pi ^{-10}r^{2}+\frac{4807642593557989}{251174796724662270}\pi ^{-10}r^{3}+\frac{6802650744626269}{2679197831729731000}\pi ^{-10}r^{4}+\frac{8930659439099103}{2.0576239347684333e+21}\pi ^{-10}r^{5}+\frac{8428983767926343}{81380634138790580000}\pi ^{-11}-\frac{35}{18432}\pi ^{-11}r-\frac{35}{3072}\pi ^{-11}r^{2}-\frac{6410190124743985}{2009398373797298200}\pi ^{-11}r^{3}-\frac{8245637266213661}{11691045083911553000}\pi ^{-11}r^{4}-\frac{1860554049812313}{514405983692108330000}\pi ^{-11}r^{5}+\frac{1324554592102711}{40690317069395290000}\pi ^{-12}r+\frac{1}{2048}\pi ^{-12}r^{2}+\frac{7849212397645697}{24112780485567580000}\pi ^{-12}r^{3}+\frac{5232808265097131}{48225560971135160000}\pi ^{-12}r^{4}+\frac{4579052908561523}{1.71023288084649e+21}\pi ^{-12}r^{5}+\frac{613012211240625}{2.736372609354384e+22}\pi ^{-12}r^{6}\\ \sum _{n=1}^\infty \frac {n^{12}}{\sinh ^{14}\pi n}&=\frac{7025868439850693}{9448126034203152}\pi ^{-7}r^{3}-\frac{8981401822275804}{3488538843398087}\pi ^{-8}r^{3}-\frac{748450151856317}{3924606198822848}\pi ^{-8}r^{4}+\frac{1331095602434083}{981151549705712}\pi ^{-9}r^{2}+\frac{2464991856359413}{981151549705712}\pi ^{-9}r^{3}+\frac{2330537755103445}{7849212397645696}\pi ^{-9}r^{4}-\frac{5244561643036313}{4415181973675704}\pi ^{-10}r^{2}-\frac{4994820612415537}{5045922255629376}\pi ^{-10}r^{3}-\frac{3729466057270267}{23547637192937090}\pi ^{-10}r^{4}-\frac{657536715331397}{548017738308354050}\pi ^{-10}r^{5}+\frac{8499225299325731}{282571646315245060}\pi ^{-11}r+\frac{5666150199550487}{15698424795291392}\pi ^{-11}r^{2}+\frac{5666150199550487}{31396849590582784}\pi ^{-11}r^{3}+\frac{457870723195999}{11417036214757376}\pi ^{-11}r^{4}+\frac{7953868562947639}{16075186990378385000}\pi ^{-11}r^{5}+\frac{4346194755337021}{15258868901023234000}\pi ^{-12}-\frac{1770671937359527}{282571646315245060}\pi ^{-12}r-\frac{3320009882549113}{70642911578811260}\pi ^{-12}r^{2}-\frac{1239470356151669}{70642911578811260}\pi ^{-12}r^{3}-\frac{182688374489475}{31396849590582784}\pi ^{-12}r^{4}-\frac{479674090967237}{8037593495189193000}\pi ^{-12}r^{5}-\frac{8640702963803677}{1.0582065950237657e+22}\pi ^{-12}r^{6}-\frac{4584996664970923}{488283804832743500000}\pi ^{-13}+\frac{1}{4096}\pi ^{-13}r+\frac{55}{24576}\pi ^{-13}r^{2}+\frac{2428350085521637}{2260573170521960400}\pi ^{-13}r^{3}+\frac{429504096758929}{861170731627413500}\pi ^{-13}r^{4}+\frac{1644596883316241}{192902243884540620000}\pi ^{-13}r^{5}+\frac{720058580316973}{2.0576239347684333e+21}\pi ^{-13}r^{6}\\ \sum _{n=1}^\infty \frac {n^{14}}{\sinh ^{14}\pi n}&=-\frac{16059127862515870}{4651385124530783}\pi ^{-6}r^{2}+\frac{5353042620838623}{41862466120777050}\pi ^{-6}r^{3}-\frac{41258314621079470}{3270505165685707}\pi ^{-9}r^{3}-\frac{42661658655810060}{5232808265097131}\pi ^{-9}r^{4}+\frac{5243709948982751}{280329014201632}\pi ^{-10}r^{3}+\frac{3495806632655167}{420493521302448}\pi ^{-10}r^{4}+\frac{1918211844585143}{11773818596468544}\pi ^{-10}r^{5}-\frac{2272643378649069}{981151549705712}\pi ^{-11}r^{2}-\frac{1657135796931613}{183965915569821}\pi ^{-11}r^{3}-\frac{1627788645864347}{458720205057216}\pi ^{-11}r^{4}-\frac{766377021238303}{5886909298234272}\pi ^{-11}r^{5}+\frac{1726405138925539}{1471727324558568}\pi ^{-12}r^{2}+\frac{651034934544311}{356782381711168}\pi ^{-12}r^{3}+\frac{6445245851988679}{8830363947351408}\pi ^{-12}r^{4}+\frac{8333247162167181}{251174796724662270}\pi ^{-12}r^{5}+\frac{367882278031071}{1461380635488944000}\pi ^{-12}r^{6}-\frac{2471898267097931}{238419826578488030}\pi ^{-13}r-\frac{5664766862099425}{29802478322311004}\pi ^{-13}r^{2}-\frac{8632025694627695}{52982183684108450}\pi ^{-13}r^{3}-\frac{651034934544311}{8562777161068032}\pi ^{-13}r^{4}-\frac{504884234952731}{161469512180140030}\pi ^{-13}r^{5}-\frac{1081447041453407}{13778731706038616000}\pi ^{-13}r^{6}-\frac{1}{32768}\pi ^{-14}+\frac{2648462429033497}{2861037918941856300}\pi ^{-14}r+\frac{2427757226614039}{238419826578488030}\pi ^{-14}r^{2}+\frac{109613024693685}{17941056908904448}\pi ^{-14}r^{3}+\frac{474989773672635}{125587398362331140}\pi ^{-14}r^{4}+\frac{12179224965965}{125587398362331140}\pi ^{-14}r^{5}+\frac{1727718683436331}{217015024370108200000}\pi ^{-14}r^{6}+\frac{4475648294971875}{4.3781961749670145e+23}\pi ^{-14}r^{7}\\ \end {align*} $$